Refining Social Graph Connectivity via Shortcut Edge Addition
MANOS PAPAGELIS, University of Toronto
Small changes on the structure of a graph can have a dramatic effect on its connectivity. While in the
traditional graph theory, the focus is on well-defined properties of graph connectivity, such as biconnectivity,
in the context of a social graph, connectivity is typically manifested by its ability to carry on social processes.
In this paper, we consider the problem of adding a small set of nonexisting edges (shortcuts) in a social
graph with the main objective of minimizing its characteristic path length. This property determines the
average distance between pairs of vertices and essentially controls how broadly information can propagate
through a network. We formally define the problem of interest, characterize its hardness and propose a
novel method, path screening, which quickly identifies important shortcuts to guide the augmentation of
the graph. We devise a sampling-based variant of our method that can scale up the computation in larger
graphs. The claims of our methods are formally validated. Through experiments on real and synthetic data,
we demonstrate that our methods are a multitude of times faster than standard approaches, their accuracy
outperforms sensible baselines and they can ease the spread of information in a network, for a varying range
of conditions.
Categories and Subject Descriptors: H.3.4 [Information Storage and Retrieval]: Systems and Software—
Information networks; H.2.8 [Database Management]: Database Applications—Data mining; J.4
[Computer Applications]: Social and Behavioral Science
General Terms: Algorithms, Performance, Human Factors
Additional Key Words and Phrases: Graph augmentation, social networks, network engineering, shortcuts,
propagation, mixing time, conductance
ACM Reference Format:
Manos Papagelis. 2015. Refining social graph connectivity via shortcut edge addition. ACM Trans. Knowl.
Discov. Data 10, 2, Article 12 (October 2015), 35 pages.
DOI: http://dx.doi.org/10.1145/2757281
1. INTRODUCTION
Group formation and segregation, the rise and fall of fashion fads, the adoption or
rejection of an innovation are all manifestations of fundamental social processes, such
as homophily, influence, and contagion, that now take place in online social networks
and can be monitored and examined through traces of online social interactions. The
ability of a social network to carry on such social processes is a characteristic that
depends, to a great extent, on the topology of the network. Consider the process of
information propagation; as individuals become aware of new or interesting pieces
of information they have the chance to pass them forward to their friends, and so
on and so forth. The number of nodes in the network that can be reached during
the propagation depends on contextual properties of the propagation itself (e.g., the
content of the information being transmitted, how susceptible nodes are in receiving
Author’s address: M. Papagelis, Computer Science Department, University of Toronto, 40 St. George Street,
Toronto ON M5S 2E4, Canada; email: [email protected].
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DOI: http://dx.doi.org/10.1145/2757281
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M. Papagelis
the information, how willing they are to further transmit it, and so on), but there is
also a genuine connection between the propagation process and the topology of the
network. It becomes obvious that slight modifications in the network topology, might
have a dramatic effect on its connectivity and thus to its capacity to carry on such social
processes.
In this paper, we approach network modification as a graph augmentation problem,
in which we ask to find a small set of nonexisting edges (from now on we refer to
them as shortcut edges or simply shortcuts) to add to the network that will optimize
a connectivity property. While in the traditional graph theory, network connectivity
typically refers to a well-defined property, such as biconnectivity, in the context of a
social graph, connectivity is usually manifested by the ability of a network to bring
users closer to each other and drive up user engagement. Having plenty of choices
for measuring the level of connectivity in the social graph, we focus on a structural
feature and try to minimize the characteristic path length of the network. This property
determines the average distance between pairs of vertices and essentially controls the
evolution of social processes in the network; for example, it controls how efficiently
information can propagate through a network. In the augmented network, information
propagates easier and faster, as it has to travel from a node to another by traversing
fewer edges of a more connected network. It is also interesting to note that while in
traditional graph augmentation problems the addition of edges can occur anywhere
in the network, in the context of a social graph, augmentation might need to respect
constraints and norms of social activity. For example, it might make sense to connect
two social peers (nodes) if only they share a contact (an adjacent node); we further
elaborate on this issue when we formally define the problem.
There are many possible application scenarios of this work. For example, suggesting
friends in a social network with the global objective of enabling efficient information
dissemination or performing faster network simulations by reducing the convergence
time of random walk processes or boosting collaboration in scientific networks by connecting the right peers. In the next section, we provide more details about a few of
these applications to further motivate our research.
1.1. Motivating Applications
We describe two applications, where a slightly modified network topology is preferred,
to motivate the problem of interest.
1.1.1. Boosting Information Propagation Processes. A longstanding interesting topic of research in network analysis has been to identify community-like structure in complex
networks (such as social, biological, or information networks). Community structure
(typically referred to as a module or cluster) refers to the occurrence of groups of nodes
in a network that are more densely connected internally than with the rest of the
network. Identifying communities is interesting because communities are often interpreted as organizational units in social networks. However, there are cases where
evidence of tightly knit communities in a network topology might constrain the evolution of processes that take place or unfold in it. This is the case of the process by which
information flows in a social network. A critical problem of this social process’ evolution
is that while information spreads effectively among members of the same community
(intracluster), it can eventually stall when it tries to break into other communities
(intercluster). In fact, as Easley and Kleinberg [2010] state:
Clusters and cascades (the outcome of information spread) can be seen as natural
opposites, in the sense that clusters block the spread of cascades, and whenever a
cascade comes to a stop, there is a cluster that can be used to explain why.
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Refining Social Graph Connectivity via Shortcut Edge Addition
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In this research, we are interested in exploring how small changes in a network’s
topology, manifested by the addition of a few shortcuts, might alleviate its community
structure, giving rise to more efficient information propagation processes.
1.1.2. Boosting Random Walks on Graphs. Graph sampling has been a common strategy to
efficiently compute interesting metrics in a large graph [Leskovec and Faloutsos 2006]
or a large social network [Papagelis et al. 2013]. The objective of sampling is to randomly
select elements (nodes or edges) according to a predetermined probability distribution,
and then to generate aggregate estimations based on the retrieved samples. Many of
the graph sampling techniques are built upon the idea of performing random walks over
the graph. One of the most popular schemes is the simple random walk (SRW) [Lovász
1993], in which a walk starts from an arbitrary node and in each step picks the next
node to visit uniformly at random from the adjacent nodes. The last node to be visited
by the walk forms a node of the sample. If the random walk is sufficiently long, then the
probability of each node to be in the sample tends to reach a stationary (probability)
distribution proportional to the degree of each node (the number of connections it has
to other nodes). Based on the retrieved samples and knowledge of such a stationary
distribution, one can generate unbiased estimations of aggregates over all nodes in the
network.
A critical problem of that sampling method is that a random walk might require to
take a very large number of steps in order to retrieve a single sample. The minimal
length of a random walk in order to reach the stationary distribution usually appears
in the literature as the mixing time or mixing rate of a random walk or Markov chain
(i.e., how fast the random walk converges to its stationary distribution). Studies on the
graph theory suggest that the mixing time is tightly related to the connectivity of the
graph. In particular, strongly connected graphs are known to have small mixing time
(fast convergence), while weakly connected graphs are known to have large mixing
time (slow convergence). How “well-knit” a graph is, is usually determined by the
conductance of a graph. It is known that graphs with large conductance have a small
mixing time. Therefore, the smaller the conductance of a graph is, the longer the
random walk of the sampling method will be.
In this research, we are interested in exploring how small changes in a network’s
topology, manifested by the addition of a few shortcuts, might increase the conductance of the graph and dramatically improve the performance of graph sampling-based
simulations, driven by faster random walks.
1.2. Contributions
The paper makes the following contributions.
—We formalize the problem of finding a set of shortcuts to add in a social graph that
will shrink the network as much as possible and formally characterize its hardness.
—We propose a novel method (path screening) for efficiently evaluating the utility of
adding shortcuts in a network.
—We present a sampling-based variant of the proposed method that improves its
performance and extents its applicability to larger graphs.
—We formally validate the claims of our methods.
—We evaluate the accuracy and efficiency of our methods using real and synthetic
data and show that they (i) outperform sensible baselines, (ii) ease the spread of
information in networks, and (iii) speed up the convergence of random walk-based
simulations in graphs, by a multitude of times and for a varying range of conditions.
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M. Papagelis
1.3. Paper Organization
The rest of the paper is organized as follows. Section 2 introduces notation and presents
background on the graph theory that is required to understand the paper. Section 3
formally defines the problem of interest and characterizes its hardness. Some theoretical results of the problem are presented in Section 4. In particular, we show that in a
graph, the sum of the lengths of all-pairs shortest paths is equal to the sum of the edge
betweenness of all its edges. We also show that the function of adding shortcuts in a
graph in order to minimize its characteristic path length, while being monotonic, it is
not submodular; therefore efficient techniques that rely on the submodularity assumption cannot be employed to solve the optimization problem. Section 5 presents efficient
heuristic methods for fast assessment of the importance of a shortcut in a network.
We experimentally evaluate our methods in Section 6. In Section 7, we review related
work and conclude in Section 8.
2. PRELIMINARIES
2.1. Network Model
Let G(V, E) be a connected undirected simple graph with V the set of vertices and E
the set of edges, where |V | = n and |E| = m. Let d(u, v) be the length of the shortest
path between the vertices u and v. The sum of all-pairs shortest path lengths is L =
(u,v)∈V ×V d(u, v). Then, the characteristic path length, L̄, is defined as the average
shortest path length over all pairs of vertices in the graph ( L̄ = L/( 2n )). L̄ is a measure
of the efficiency of information transport on a network. It is also a measure of how
close users are to each other, which encourages social interaction and drives up user
engagement. The range of an edge R(u, v) is the length of a shortest path between u
and v in the absence of that edge. An edge with R(u, v) ≥ 2 is called a shortcut.
2.2. SRW on a Graph
We recall a few definitions of random walks on graphs. For more detailed exposition,
see [Sinclair 1992]. We define the degree of a node v ∈ V as the number of nodes in V
adjacent to v and denote it by deg(v). We also define the neighborhood of a node v to be
N(v) = {u : d(u, v) = 1} (so v ∈
/ N(v)). A SRW on a graph G is a Markov Chain Monte
Carlo method that takes successive random steps according to a transition probability
matrix P = [ pvu] of size n × n where the (v, u)th entry in P is the probability of moving
from node v to node u defined as
1
, if u ∈ N(v)
pvu = deg(v)
0,
otherwise.
At any given iteration t of the random walk, let us denote with π (t) the probability
distribution of the random walk state at that iteration (π (t) is a vector of n entries).
The state distribution after t iterations is given by π (t) = π (0) · P t , where π (0) is the
initial state distribution. The probability of a t steps random walk starting from v
t
and ending at u is given by Pvu
. For irreducible and aperiodic graphs (which is the
case of undirected connected social graphs), the corresponding Markov chain is said
to be ergodic. In that case, the stationary distribution π of the Markov chain (the
distribution after a random walk of length μ, as μ → ∞.) is a probability distribution
that is invariant to the transition matrix P (i.e., satisfies π = π · P). For an undirected
unweighted connected graph G, the stationary distribution of the Markov chain over
G is the probability vector π = [πv ], where πv = deg(v)/2|E|.
2.2.1. Mixing Time. The mixing time of a Markov process is a measure of the minimal
length of the random walk in order to reach the stationary distribution. Let (t) be
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the relative point-wise distance between the current sampling distribution and the
stationary distribution, after t steps of SRW:
t
|Pvu − πu|
(t) =
max
πu
v,u∈V,u∈N(v)
t
where Pvu
is the element of P t with indices v and u.
Definition 2.1 (Mixing Time of a Graph). The mixing time of a graph is the minimum
number of steps t required by a random walk to converge; that is, (t) ≤ , where is
a threshold.
Note that social graphs have been shown to have relatively larger mixing time than
what it was usually anticipated [Mohaisen et al. 2010].
2.3. Community Structure and Conductance
A common characteristic in the study of complex networks is community structure. The
quality of a community is commonly defined using modularity or flow-based measures.
We adopt a more intuitive community quality concept, conductance.
Definition 2.2 (Conductance of a Set of Nodes). Let G = (V, E) be an undirected
graph and C ⊂ V be a set of nodes (C can be thought of as a cluster or community).
Then, the conductance (C, C̄) (or simply (C)) is given by:
(C) = (C, C̄) =
|EC,C̄ |
Vol(C)
where
C̄ = V − C
EC,C̄ = {(v, u) ∈ E|v ∈ C, u ∈ C̄}
deg(v).
Vol(C) =
v∈C
The notion of conductance (C, C̄) of a set of nodes C may be thought of as the ratio
between the number of connections pointing outside the community C (to nodes in C̄)
and the number of connections inside C. It is widely used to capture quantitatively the
notion of a good network community as a set of nodes that has better internal connectivity than external connectivity [Leskovec et al. 2008]. Note that more community-like
sets of nodes have lower conductance (Figure 1).
Using conductance as a measure of network community quality, one can define the
conductance of a graph.
Definition 2.3 (Conductance of a Graph). Let G = (V, E) be an undirected graph.
The conductance G of G is given by:
G = min (C, C̄).
|C|≤|V |/2
2.3.1. Graph Partitioning. The conductance G of a graph is intractable to compute exactly (NP-hard), since we need to search over all possible cuts of a graph (of any
community size |C|) and find the one with minimum conductance. Therefore, we have
to resort to efficient approximation methods for computing partitions of graphs with
minimum conductance over all the possible cuts. This is the graph partitioning problem, in which the goal is to find groups of nodes (clusters) in a graph G, with roughly
equal size, such that the number of edges between the groups is minimized and a merit
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M. Papagelis
Fig. 1. Community structure in complex networks. For example, (blue) = 3/13, (green) = 2/18, (red) =
3/25. The set of green nodes (that has the lower conductance) forms a more community-like set of nodes than
the set of blue or red nodes. The conductance of the graph is G = 2/18, which represents the minimum
conductance of any set of nodes C, where |C| ≤ |V |/2.
function is optimized. In our analysis, the popular graph partitioning package Metis
[Karypis and Kumar 1998] has been used to obtain sets of nodes that have very good
conductance scores. Metis is a multilevel graph partitioning algorithm that can give
both fast execution times and high quality results.
2.4. Relation between Conductance and Mixing Time
Conductance is a measure of the tendency of a random walk to move out of a subset
of nodes. The relationship between the graph conductance and the mixing time of a
random walk is explained by the following inequalities [Alon 1986; Jerrum and Sinclair
1989]:
t
(G)2
2|E|
t
(1 − 2(G)) ≤ (t) ≤
1−
.
minv∈V · deg(v)
2
The aforementioned inequalities effectively provide a characterization of a SRW’s mixing time in terms of the graph conductance (G). One can see that the larger the graph
conductance (G) is, the smaller the mixing time t will be. Intuitively, high conductance would imply that the graph has a relatively low mixing time (see [King 2003] for
details and discussion). Note, as well, that because of the logscale relationship between
(G) and the mixing time t, a small change on (G) will lead to a significant change of
t. Now, if is a threshold, let
2|E|
(1 − (G)2 /2)t ≤ .
minv∈V · deg(v)
By applying the natural logarithm to both sides, we get
1
log
t≥
2|E|
2
log(1 − (G) /2)
.
(1)
minv∈V ·deg(v)
Based on the minimum value of t in (1), we can define a theoretical mixing time of
the fastest random walk in a graph. We employ this measure in the experimental
evaluation.
3. THE PROBLEM
Let G(V, E) be a connected undirected simple graph. Now, assume that we are allowed
to augment G with a small number of edges, with the objective of optimizing a connectivity property. We denote Gaug an augmentation of G (see Figure 2). While having
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Refining Social Graph Connectivity via Shortcut Edge Addition
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Fig. 2. Graph augmentation of G to Gaug . (a) The input consists of G, an integer k and an augmentation
policy P, (b) the augmented graph Gaug is a possible output. We ask to determine the set of shortcuts S that
when used to augment G, u(S) is maximized.
plenty of choices for measuring the level of connectivity in Gaug , we focus on a structural
feature, namely the characteristic path length, L̄. Note that, adding edges in a network
results in altering some of the st-shortest paths in the graph, where s, t ∈ V . It is worth
noting immediately that while adding long-haul edges (shortcuts with large range R)
may bring a large gain in terms of connectivity for some (s, t) pairs (the new length of
a shortest path connecting them will be much shorter), long-haul edges are likely to
affect the shortest path length of only a few (s, t) pairs. On the other hand, suggesting
short-haul, intracommunity edges (shortcuts with small range R) is likely to affect the
shortest path length of many (s, t) pairs, but the gain in connectivity for each of them
might be limited (i.e., the new length of a shortest path connecting them will not be
much shorter). Therefore, in trying to suggest edges to be added to the graph we are
facing an interesting tradeoff. Let CS represent the set of all candidate shortcuts for
addition. This set represents all shortcut edges (i, j) between any nonadjacent nodes
i, j ∈ G (i.e., R(i, j) ≥ 2). Note that in a sparse network of n nodes (most real networks
are sparse), there exist (almost) n(n − 1)/2 candidate shortcuts. Now, assume the existence of a utility set function u : 2CS → R that maps a given subset of shortcut edges
S ⊆ CS to a nonnegative real number that represents the difference of the sum of
all-pairs shortest path length (L(.)) in G = (V, E) and Gaug = (V, E ∪ S). Formally,
u(S) = L(G) − L(Gaug ).
(2)
Another critical issue is related to the semantics of the set of shortcut edges S. From a
network optimization perspective, one might be interested to add shortcuts anywhere
in the network. However, in some settings (e.g., suggesting friends in a social network),
it might be required to suggest a few shortcuts to each node in the network, in a
balanced way. In both scenarios, we are interested in a more efficient network, but we
are required to respect an augmentation policy P of adding shortcuts in the network.
We consider two particular cases that we see to be of more interest in practice and
define the following augmentation policies.
—Network-Centric Policy: Graph augmentations can be applied anywhere in the network. In this case, an integer k that is provided as input represents the total number
of shortcuts to be added, not necessarily equally distributed over nodes. Therefore,
|S| ≤ k.
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M. Papagelis
—Node-Centric Policy: Graph augmentations should be applied equally over all nodes
in the network. In this case, an integer k that is provided as input represents the
total number of shortcuts to be attached to each node. Therefore, |S| ≤ k|N|.
The two augmentation policies control the level of flexibility in suggesting specific
shortcuts to add in a graph. For example, following the network-centric policy, it is
allowed to select k shortcuts and attach them to a single node, in an attempt to resemble
the topology of a star network. The intuition behind this is that stars are network
topologies with very small average shortest path distance [Meyerson and Tagiku 2009].
On the other hand, the node-centric policy restricts the number of shortcuts that can be
attached to a single node and, in addition, requires that the same number of shortcuts
is attached to any node in the network (e.g, one shortcut to each node). It becomes
clear that the augmentation policy can have important implications on the way that
algorithms that try to solve the problem operate. In one case, we seek to find the best k
shortcuts globally in the network, therefore early stopping conditions can be considered,
as soon as, the required number of shortcuts has been reached. In the other case, we
seek to find the best k shortcuts for each node, therefore we might need to evaluate a
possible larger set of candidate shortcuts before meeting the requirement. We further
elaborate on this issue in Section 6.
We are now in position to formally define the problem of interest in this paper.
PROBLEM 1. Let G = (V, E) be a connected, undirected, simple graph, and let CS =
{(i, j) : R(i, j) ≥ 2} be a set of candidate shortcuts for augmentation. If Gaug = (V, E ∪
S) is the augmented graph of G, then the objective is, given G, an integer k and an
augmentation policy P, to find a set of shortcut edges S ⊆ CS that maximizes u(S).
It is easy to see that maximizing u(S) is equivalent to maximizing u
(S) = L̄(G)− L̄(Gaug ),
since we only add new edges in G and do not change its number of nodes.
3.1. Problem Hardness
We establish the hardness of our problem by exploiting its equivalence with the
weighted version of the Maximum Coverage Problem (WMCP). Not surprisingly, it
turns out that our problem is NP-hard.
THEOREM 3.1. Problem 1 is NP-Hard.
PROOF. We prove the claim by reducing our problem to the WMCP. In WMCP the
input is a universe U of n elements
U = {e1 , e2 , . . . , en}, a collection C = {C1 , C2 , . . . , Cm}
of m subsets of U, such that i Ci = U, a weight function w : C → R that assigns
non-negative weights to subsets and an integer k ≤ m. The goal is to select a subset
S ⊆ C, such that |S| ≤ k and the combined weight in the union of the subsets to be
maximized. It is easy to see that our problem can be reduced to WMCP as follows:
let a universe U of n elements, where each element represents an (s, t) pair of the
graph G. Let, as well, a collection of candidate shortcuts CS = {CS1 , CS2 , . . . , CSm},
where each candidate shortcut CSi represents a set of (s, t) pairs; this is the set of
(s, t) pairs whose shortest path lengths are affected (shortened) by the addition of the
corresponding shortcut in G. Moreover, every candidate shortcut is assigned a weight w
that represents the utility u of adding the corresponding shortcut in G. Then, given an
integer k, a collection of candidate shortcuts and their utility scores, the goal is to select
a subset of shortcuts S ⊆ CS, such that |S| ≤ k that when added in G the combined
utility is maximized. Problem 1 is NP-hard by reduction to the WMCP, one of the
classical NP-hard combinatorial optimization problems. This concludes our proof.
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3.2. Negative Results
The maximum coverage problem is NP-hard, and cannot be approximated within 1 −
1
+ o(1) ≈ 0.632 under standard assumptions. This result matches the approximation
e
ratio achieved by a generic greedy algorithm used for maximization of submodular
functions with a cardinality constraint [Nemhauser et al. 1978]. The generic greedy
algorithm for maximum coverage chooses sets according to one rule: at each stage,
choose a set which maximizes the number of elements covered by the new set but not
by any of the preceding ones. Moreover, due to Feige, we know that the greedy algorithm
is essentially the best-possible polynomial time approximation algorithm for maximum
coverage [Feige 1998]. That inapproximability result applies to the weighted version of
the maximum coverage problem, as well, since it holds the maximum coverage problem
as a special case.
These results, suggest that the greedy heuristic that adds the current best shortcut
to a graph G, until k shortcuts are added would be the best possible heuristic. However,
we show that the function u(.) of adding shortcuts in a graph in order to minimize the
average shortest path length, while being strictly monotonic (see Theorem 4.5), it is not
submodular (see Theorem 4.6). Submodular functions have some nice parametric and
postoptimality properties and related optimization problems can be solved efficiently
if the submodularity property is valid [Filmus and Ward 2012; Kapralov et al. 2013;
Nemhauser et al. 1978; Vondrak 2008]. Since u(.) does not possess the submodularity
property, techniques that utilize it for approximation cannot be used. Thus, strictly
speaking, we cannot expect to have an efficient algorithm with a provable approximation guarantee for our problem. Nevertheless, given the hardness of the problem, a
greedy algorithm of successively adding shortcuts by repeatedly picking the best shortcut could still be a good heuristic (see Section 5). But, since the search space is massive,
the greedy algorithm would be extremely slow. So, we focus our attention on designing an efficient heuristic algorithm, by avoiding unnecessary coverage evaluations. We
explain the details of our algorithm in Section 5.
4. SOME THEORETICAL RESULTS
In this Section, some theoretical results are discussed.
4.1. Relationship between Characteristic Path Length and Edge Betweenness in G
We have structured the problem statement around minimizing the characteristic path
length of a graph, L̄. We discuss here the relationship between L̄ and another popular
network structural property, namely edge betweenness.
Definition 4.1 (Edge Betweenness [Girvan and Newman 2002]). The betweenness
centrality of an edge or simply edge betweenness is the number of shortest paths
between pairs of nodes that run along it. If there is more than one shortest path
between a pair of nodes, each path is assigned equal weight such that the total weight
of all of the paths is equal to unity. Formally, let σst = σts denote the number of shortest
paths from s ∈ V to t ∈ V and let σst (e) denote the number of shortest paths from s to t
that pass through edge e. Then, the edge betweenness of e is given by
σst (e)
.
be =
σst
s,t∈V,s=t
Definition 4.2 (Total Betweenness). Let B be the sum of the edge betweenness of all
edges in G, given by
B(G) =
be .
e∈E
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Interestingly, we show that in a graph G, the sum of all-pairs shortest paths, L(G),
is equal to the sum of the edge betweenness of all edges, B(G).
THEOREM 4.3. Let G(V, E) be a connected undirected simple graph with V the set of
vertices and E the set of edges. If L is the length of all-pairs shortest paths and B is the
sum of the edge betweenness of all edges in G, then B(G) = L(G).
PROOF. Without loss of generality, we fix a pair of nodes s, t ∈ V . Assume that there
are pst such shortest paths between s, t, all of which have the same length d(s, t). Note
that each path pi ∈ pst consists of |d(s, t)| edges and contributes to the edge betweenness
of each edge along the path by a factor of p1st . So, each single shortest path pi contributes
d(s,t)
, and all pst shortest paths of the
pst
d(s,t)
). Now, considering and summing
pst
to the total edge betweenness by a factor of
nodes s, t ∈ V contribute by d(s, t) (= p ·
total edge betweennesses of all s, t pairs in G we get B =
st
s∈V
t∈V
pair of
up the
d(s, t) or B = L.
Due to Theorem 4.3, minimizing u(S) in equation (2) is equivalent to minimizing:
u
(S) = B(G) − B(Gaug ).
(3)
4.2. Properties of the Utility Function u (.)
We show some properties of the utility function u(.).
LEMMA 4.4. Let G(V, E) be an undirected graph and let Gaug = (V, E ∪ {e}) be an
augmentation of G after the addition of an edge e ∈ CS. Then, L(Gaug ) < L(G).
PROOF. Without loss of generality, we fix a pair of nodes s, t ∈ V and let d(s, t), d
(s, t)
be the length of a shortest path connecting them in G and in Gaug , respectively. For any
shortest path between s and t in Gaug , there are two options:
—it does not traverse e: then d
(s, t) = d(s, t);
—it traverses e: then d
(s, t) < d(s, t).
Therefore, for any pair of nodes s, t ∈ V it holds that d
(s, t) ≤ d(s, t). Summing up the
shortest path lenghts of all s, t pairs in Gaug and G we get L(Gaug ) ≤ L(G) (monotonically
decreasing). Now, let the new edge e be incident to two nodes, say x and y. It is easy
to see that the (x, y)-shortest path in Gaug is always shorter than the (x, y)-shortest
path in G. Therefore, after the addition of the edge e in G, there is at least one pair of
vertices, where d
(x, y) < d(x, y). It follows that L(Gaug ) < L(G).
THEOREM 4.5. u(.) is strictly monotonic.
PROOF. From Lemma 4.4, it follows that adding an edge e ∈ CS to a set S ⊂ CS
will always cause u(.) to increase, so u(S ∪ {e}) > u(S) for any edge e and set S (strictly
increasing).
THEOREM 4.6. u(.) is not submodular.
PROOF. Let Z be a finite set. A function f : 2 Z → R is submodular if for any X ⊂ Y ⊂ Z
and e ∈ Z \ Y , it holds that:
f (X ∪ {e}) − f (X) ≥ f (Y ∪ {e}) − f (Y ).
Intuitively, a submodular function over the subsets demonstrates “diminishing returns.” We provide a counterexample that proves that the utility function u(.) does not
possess the submodularity property.
Counterexample: Let G(V, E) be an undirected graph and let the sets X = {}, Y = {e1 }
and Z = {e1 , e2 } (see Figure 3). It is easy to see that X ⊂ Y ⊂ Z. Note that the edges
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Fig. 3. Submodularity counterexample. Assume G(V, E) and let the sets X = {}, Y = {e1 } and Z = {e1 , e2 }.
in G form two very large fully connected subgraphs, each of size n (i.e., the number of
nodes in each subgraph). Then, for e = e2 ∈ Z \ Y holds that:
u(X ∪ {e}) − u(X) = (L(G) − L(G(V, E ∪ X ∪ {e})))
− L(G) − L(G(V, E ∪ X))
= L(G(V, E ∪ X)) − L(G(V, E ∪ X ∪ {e}))
n(n − 1)
n(n − 1)
5
−5
=0
2
2
and
u(Y ∪ {e}) − u(Y ) = (L(G) − L(G(V, E ∪ Y ∪ {e})))
− (L(G) − L(G(V, E ∪ Y )))
= L(G(V, E ∪ Y )) − L(G(V, E ∪ Y ∪ {e}))
n(n − 1)
n(n − 1)
n(n − 1)
5
−4
=
.
2
2
2
The aforementioned coefficients 4 and 5 represent the typical shortest path lengths for
(almost) all pairs of nodes in the network instances.
Asymptotically it holds that:
u(X ∪ {e}) − u(X) < u(Y ∪ {e}) − u(Y ).
Therefore, we obtain a contradiction and we can conclude that the submodularity
property does not hold.
5. SHORTCUT UTILITY COMPUTATION
Let uxy represent a measure of the utility of adding a single shortcut edge (x, y) ∈ CS
in G. According to (2), it is:
uxy = L(G) − L(Gaug )
To evaluate uxy , we need to recompute the shortest paths between all pairs of vertices
s, t ∈ V in Gaug . This is because a previously found shortest path connecting s, t may
now need to traverse the newly added edge (x, y). Let d
(s, t) represent the length of the
shortest path between s and t in Gaug . Then:
(d(s, t) − d
(s, t)).
uxy =
(s,t)∈V ×V \E
In the following sections, we describe methods for computing the utility of all candidate
shortcuts for augmentation. We first describe a greedy method. Then, we explain why
this approach is not scalable and thus not suitable for larger graphs, and discuss
methods that can scale up its performance.
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M. Papagelis
ALGORITHM 1: Greedy Method
Input: A Social Graph G(V, E) and a set of Candidate Shortcuts CS = {(x, y) : R(x, y) ≥ 2}
Output: A shortcut edge emax with maximum utility
L(G) = computeL(G);
umax = 0;
forall the (x, y) ∈ CS do
Gaug ← G(V, E ∪ {(x, y)}) ;
L(Gaug ) = computeL(Gaug );
uxy = L(G) − L(Gaug );
if uxy > umax then
umax = uxy ;
emax ← (x, y);
end
end
return emax ;
// add (x, y) in G
5.1. Greedy Method
We describe a general greedy method (see Algorithm 1), where all candidate shortcuts
are considered and a shortcut is selected in each iteration that lowers the average
shortest path length of the graph as much as possible (locally optimal choice). The
algorithm operates in k rounds. In each round i, the algorithm determines the shortcut
with the highest utility and adds it to the set of the top-k edges to be suggested for
augmentation. Equivalently, this means that the shortcut selected in round i is the
one that minimizes the average shortest path length in this round and for this graph.
In more detail, the method (see Algorithm 1) iterates over all candidate shortcuts CS
and for each shortcut (x, y) ∈ CS estimates its associated utility uxy by adding it to
the graph and computing the difference between the sums of all-pairs shortest path
lengths in the original graph G and the augmented graph Gaug . Then, the candidate
shortcut (x, y) is removed from the graph and the process is repeated for the rest ones.
In order to determine the best shortcut for augmentation, the method needs to compute
the utility of all candidate shortcuts of a graph.
The worst-case time complexity of the greedy method is O(k · n2 · n(nlogn + m)): the
utility computation of a single candidate shortcut has time complexity O(n(nlogn + m))
(equivalent to running the Johnshon’s algorithm for all-pairs shortest paths one time
[Johnson 1977], using Fibonacci heaps [Fredman and Tarjan 1987] in the implementation of Dijkstra’s algorithm), there are |CS| = n(n − 1)/2 − m = ∼n2 candidate
shortcuts that need to be evaluated in order to pick the best one, and to find the
best k, one would need to repeat the procedure k times.
5.2. Relaxation of the Greedy Method
It is possible to drop k if instead of computing the best candidate shortcut, one at a
time, we use the already computed shortcut utility scores from the first iteration and
determine the best k shortcuts all at once (batch processing). While this variant is
expected to be faster (time complexity is O(n2 · n(nlogn+ m))), it might affect the overall
performance of the shortcuts added, as the addition of a shortcut might lower the utility
of a subsequent one. We discuss this tradeoff in the experimental evaluation section.
5.3. Path Screening Method
We describe a novel method that can speed up the process of computing the utility
of all candidate shortcuts for augmentation. The method works by decomposing the
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computation of the sum of the shortest path lengths into subproblems. It is easy to see,
that in the general case the addition of a shortcut (x, y) ∈ CS to G, alters the shortest
path lengths of many (s, t) pairs with s ∈ V and t ∈ V . We say that these (s, t) pairs
depend on shortcut (x, y). However, many st-shortest paths do not depend on (x, y), and
as such the computation of the shortcut utility does not need to consider them. Let
Dst (x, y) be the set of (s, t) pairs that depend on the shortcut (x, y). For each (s, t) pair,
it holds that:
(s, t) ∈ Dst (x, y) if d
(s, t) < d(s, t)
(s, t) ∈
/ Dst (x, y) if d
(s, t) = d(s, t).
Note that for the majority of shortcuts (x, y), the number of (s, t) pairs that actually
depend on a shortcut is much smaller than the total number of node pairs in G (i.e.,
|Dst (x, y)| |V × V \ E|). This suggests large savings in the computation of the utility
uxy , since in fact we only need to sum up the differences in the shortest path length of
a much smaller set of pairs (s, t) ∈ Dst (x, y):
(d(s, t) − d
(s, t)).
(4)
uxy =
(s,t)∈Dst (x,y)
Next, we present a theorem that states that given a shortcut edge of a fixed range
δ in G, it is more beneficial to attach this shortcut on nodes of an existing st-shortest
path. We use this theorem to further decompose uxy in (4).
THEOREM 5.1. Let G(V, E) be an undirected graph and let ps be a shortest path and
pa be an alternative (i.e., nonshortest) path between nodes s ∈ V , t ∈ V . Assume an
addition of a new shortcut {(x, y) : R(x, y) = δ, δ ≤ d(s, t)} in G. Then, it holds that
d
ps (s, t) < d
pa (s, t), if both x ∈ V , y ∈ V are nodes traversed by ps in G, where d
ps (s, t),
d
pa (s, t) are the lengths of the shortest paths ps , pa in Gaug , respectively.
PROOF. Let ps represent a shortest path in G and assume that one (or both) of x ∈ V ,
y ∈ V is not traversed by ps (or any of the shortest paths connecting s, t). Now, let pa
represent a nonshortest path in G that connects s ∈ V , t ∈ V and traverses both x, y.
Then, (since pa is not a shortest path) it holds that:
d pa (s, t) > d ps (s, t)
pa
(5)
ps
where d (s, t), d (s, t) are the lengths of the paths ps and pa in G. Now assume addition
of a shortcut {(x, y) : R(x, y) = δ} in G. Then, it holds that:
d pa (s, t) = d
pa (s, t) + (δ − 1)
(6)
pa
where d (s, t) is the length of the path connecting (s, t) in the augmented graph Gaug .
This is due to the constraint that R(x, y) = δ. By (5) and (6), it follows that:
d
pa (s, t) + (δ − 1) > d ps (s, t).
(7)
On the contrary, it is easy to see that if both x, y were traversed by ps , then after
addition of an edge {(x, y) : R(x, y) = δ} in G, it would hold that:
d ps (s, t) = d
ps (s, t) + (δ − 1).
(8)
d
pa (s, t) > d
ps (s, t).
(9)
By (7) and (8), it follows that:
This concludes our proof.
Theorem 5.1 states that given a shortcut edge of a fixed range δ in G, it is more
beneficial to attach this shortcut on nodes of an existing st-shortest path. As such, it
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M. Papagelis
SP (x, y). It doesn’t belong to
Fig. 4. The (s, t) pair belongs in both Dst (x, y) and Dst (x , y
), but only to Dst
SP (x , y
), as x or y
are not nodes of the st-shortest path.
Dst
constrains the set of (s, t) pairs that depend on a candidate shortcut (x, y) to be (s, t)
pairs that their shortest paths in G traverse both nodes x and y. Let this set of (s, t)
pairs that depend on a candidate shortcut (x, y) be DstSP (x, y) (see Figure 4). Then, we
can further decompose uxy in (4) by:
((d(s, x) + d(x, y) + d(y, t))
uxy =
(s,t)∈DstSP (x,y)
− (d(s, x) + d
(x, y) + d(y, t)))
(d(x, y) − d
(x, y)).
=
(s,t)∈DstSP (x,y)
Note that the shortest path length d(x, y) represents the range of (x, y) in G, R(x, y).
Moreover, the shortest path length d
(x, y) represents the length of the newly added
shortcut in Gaug , which is constant (d
(x, y) = 1). Therefore,
(R(x, y) − 1).
(10)
uxy =
(s,t)∈DstSP (x,y)
Eventually, the utility uxy depends on the range of (x, y) in G and the size of DstSP (x, y),
in accordance with our initial intuition.
5.3.1. Fast Assessment of Shortcut Utility Scores. We look to efficiently compute the utility
score uxy of any shortcut (x, y) ∈ CS in G according to (10). Algorithm 2 describes
the steps to solve the problem. It uses a modification of Johnson’s Algorithm [Johnson
1977] that does not only provide the lengths of the all-pairs shortest paths, but also
reconstructs and stores them in P. Then, for each shortest path p ∈ P, it determines
what are the (x, y) shortcuts that could shorten the current path p by sliding a variablesize window of size δ over the path’s nodes (see Figure 5). Note that δ takes values that
range from δ = 2 (shortcuts between nonadjacent nodes) to δ = l, where l is the length
of the current path p. It is easy to see that the maximum δ to be considered will
be equal to the diameter D of G, where D = maxi, j d(i, j) (the longest shortest path
between any two nodes in G), so the range of values of δ is 2 ≤ δ ≤ D. As we slide the
window over a path, the window’s endpoints determine the x and y nodes of a candidate
shortcut (x, y). Each time a shortcut (x, y) is encountered in a path p, an increment
equal to (δ − 1) is contributed to its utility. The procedure continues over all paths
p ∈ P and at the end, for each (x, y) its utility uxy is computed in accordance to (10).
The time complexity of our path screening method is O(n(n log n+ m)) + O(n2 L̄(G) L̄(G)):
it requires O(n(n log n + m)) time to find the shortest paths in G (equivalent to running
the Johnson’s algorithm ones). Then, for each path (O(n2 )) needs to vary the δ-size of
the window (O( L̄(G))) and slide the window over the path p (O( L̄(G))).
5.4. Path Screening Sampling-Based Method
Algorithm 2 describes an efficient way to accurately compute the utility of each candidate shortcut uxy . However, it is still not very practical for large graphs as it requires
ACM Transactions on Knowledge Discovery from Data, Vol. 10, No. 2, Article 12, Publication date: October 2015.
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Fig. 5. Path screening method. (a) The input consists of a graph G. (b) All-pairs shortest paths are extracted
from G. (c) A δ-size window (for varying δ) is slided over each shortest path. (d) Occurrences of any candidate
shortcut (x, y) found contribute to its utility uxy .
ALGORITHM 2: Path Screening Method
Input: A Social Graph G(V, E)
Output: A Map between Candidate Shortcuts and their utility scores
U ← ∅;
P = get All PairsShortest Paths(G);
forall the p ∈ P do
l = length( p);
// Variable size of window
for δ = 2 to l do
// Slide a δ-size window over nodes of p
for i = 0 to l − δ do
x = p[i] ;
y = p[i + δ] ;
if (x, y) ∈
/ U then
uxy = (δ − 1);
add < (x, y), uxy > in U ;
else
update < (x, y), uxy + (δ − 1) > in U ;
end
end
end
end
return U
// ith node of path p
// (i + δ)th node of path p
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M. Papagelis
that all-pairs shortest paths can be efficiently computed (O(n(n log n + m))) and then
screened (O(n2 L̄(G) L̄(G))). In this section, we describe a sampling-based variant of the
path screening method for efficiently computing the utilities of the candidate shortcuts.
Ideally, we would like to sample uniformly at random a few shortest paths from the
set of all available ones. Then, we would feed them to our Algorithm 2 and compute
estimates of the utility of each candidate shortcut based on that sample. However, uniform shortest path sampling is rarely feasible [Kandula and Mahajan 2009]. Instead,
we sample uniformly at random a set of Q nodes Q = {q1 , . . . , qq }. Then, starting from
a node qi , i = 1, . . . , q, where q n, we execute the Dijkstra algorithm [Dijkstra 1959]
and compute the single source shortest path tree from qi to all nodes x in the graph.
Now, instead of computing the utility of a candidate shortcut by operating over all-pairs
shortest paths, we operate only on the set Q of shortest paths returned by the sample:
Q
uxy
=
d(s, t) − d
(s, t).
s∈Q,t∈V
The way we perform the sampling reduces the search space where candidates can be
found (as not all shortest paths become available) and is likely to introduce bias. The
main reason for the bias is that some paths are more likely to exist in the shortest
path trees of sampled sources [Kandula and Mahajan 2009]. However, in our setting,
we are interested in finding only a few candidate shortcuts that have very high utility. Note that edges with high betweenness centrality [Newman 2005] in G are more
likely to be in the sample as by definition they are edges that are traversed by many
st-shortest paths. And, since important candidate shortcuts are shortcuts (x, y) that
their endpoints x, y are found in many st-shortest paths, it is expected that candidate
shortcuts with high utility will be sufficiently represented in the single-source shortest
path trees of the sampled nodes. Therefore, our path screening method will be able to
detect them. We assess the effect of the bias in the experimental evaluation. The rest of
the algorithm remains the same. The point is that the aforementioned computation is
relatively efficient, since it does not need to compute all-pairs shortest paths. Instead,
we compute q times the single-source shortest-path Dijkstra. Thus, the time complexity of our sampling-based path screening method is O(q(n log n + m)) + O(qnL̄(G) L̄(G)),
where q n.
5.4.1. Sampling-Based Method Estimation. In this section, we formally describe the
sampling-based estimation method. First, we employ a commonly used graph embedding technique, called reference node embedding [Goldberg and Harrelson 2005;
Kleinberg et al. 2004; Qiao et al. 2012], to show that the distances between any two
nodes in a graph can be estimated if a small set of reference nodes are selected [each
serving as the root of a shortest path tree (SPT)] and then the distances between these
nodes to all other nodes in a graph are computed.
Definition 5.2. Let T be a rooted tree with n nodes. The lowest common ancestor
(lca) of two nodes u and v in T is the shared ancestor of u and v that is located farthest
from the root.
THEOREM 5.3. Given a set of reference nodes Q = {q1 , q2 , . . . qq } ∀s, t ∈ V it holds that:
d(s, t) ≥ max{|d(s, q) − d(q, t)|}
(11)
d(s, t) ≤ min{d(lcaq , s) + d(lcaq , t)}
(12)
q∈Q
q∈Q
PROOF. Given a set of reference nodes Q = {q1 , q2 , . . . qq }, our sampling-based method
considers the shortest paths for the node pairs {(q, v) : q ∈ Q, v ∈ V } by computing the
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Fig. 6. Example shortest path trees (SPT). (a) SPT rooted at q ∈ Q. (b) Lowest common ancestor (lca) in a
SPT.
single-source shortest path trees for every q ∈ Q. The shortest-path distance in graphs
is a metric, and therefore it satisfies the triangle inequality. That is, given a pair of
nodes (s, t), ∀q ∈ Q the following inequalities hold (see Figure 6(a)):
d(s, t) ≥ |d(s, q) − d(q, t)|
(13)
d(s, t) ≤ d(s, q) + d(q, t)
(14)
By considering the lcas, tighter bounds are feasible [see Figure 6(b)]. ∀q ∈ Q it holds
that:
d(s, q) + d(q, t) = d(lca, s) + d(lca, t) + 2d(lca, q).
As d(lca, q) ≥ 0, we have:
d(s, lca) + d(lca, t) ≤ d(s, q) + d(q, t).
Consequently,
d(s, t) ≤ d(s, lca) + d(lca, t).
By considering all reference nodes q ∈ Q, we have tighter bounds:
d(s, t) ≥ max{|d(s, q) − d(q, t)|}
(15)
d(s, t) ≤ min{d(lcaq , s) + d(lcaq , t)}
(16)
q∈Q
q∈Q
where lcaq is the lca of s, t in an SPT rooted at q ∈ Q. This concludes the proof.
Then, we show that despite the fact that our sampling-based method picks shortcuts
to add based on a small sample of shortest paths, the addition of these shortcuts in G
will eventually shrink the shortest path length of any (s, t) pair.
THEOREM 5.4. Let di (u, v) denote the distance between u ∈ V , v ∈ V after i iterations,
where in each iteration a single shortcut is added in G. Then, ∀s, t ∈ V and sufficiently
large n it holds that:
dn(s, t) < d(s, t)
(17)
PROOF. Assume addition of a shortcut in G. Then, ∀u, v ∈ V in G
it holds that:
d
(u, v) ≤ d(u, v)
(18)
since the new shortcut might shorten the distance between u ∈ V , v ∈ V or not. Now,
let di (u, v) denote the distance between u ∈ V , v ∈ V after i iterations, where in each
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M. Papagelis
Table I. Comparison of the Worst-Case Time Complexity of the Various Methods
Method
Worst-Case Time Complexity
Greedy
Relaxation of Greedy
Path Screening
Path Screening Sampling-Based
O(k · n2 · n(nlogn + m))
O(n2 · n(nlogn + m))
O(n(n log n + m)) + O(n2 L̄(G) L̄(G))
O(q(n log n + m)) + O(qnL̄(G) L̄(G)), where q n
iteration a single shortcut is added in G. Then, it holds that:
d1 (s, t) ≤ d1 (s, q) + d1 (q, t) = α1
d2 (s, t) ≤ d2 (s, q) + d2 (q, t) = α2
...
dn(s, t) ≤ dn(s, q) + dn(q, t) = αn
and because of (18), it holds that:
α1 ≥ α2 ≥ · · · > · · · ≥ · · · > · · · ≥ αn.
Note that with a sufficient large n there will eventually be instances where the equality
can be dropped as a new edge will shrink one of the distances d(s, q) or d(q, t). Finally,
after n iterations, the distance between s and t will fall below a constant α:
dn(s, t) ≤ dn(s, q) + dn(q, t) = αn ≤ α
where α d(s, t). Therefore,
dn(s, t) < d(s, t).
This concludes the proof.
Theorem (5.3) states that for any st-shortest path not included in the sample, alternative shortest paths exist that can be used to estimate its length and this estimation
is bound. Theorem (5.4) states that by adding shortcuts that aim to shorten the length
of specific shortest paths in the sample, eventually we manage to shorten many more
st-shortest path lengths in the original graph, and thus successfully shrink the characteristic path length of the graph. The premise is that our sampling-based method
is not only fast, but it also succeeds in approximating the solution found by applying
the path screening method on all-pairs shortest paths. We experimentally validate our
claims and empirically show the tradeoff between efficiency and quality degradation
in the evaluation section.
5.5. Summary of Methods
Table I provides a summary of the worst-case time complexity of the various methods
discussed for easy reference.
6. EXPERIMENTAL EVALUATION
In this Section, we evaluate methods that suggest shortcuts for addition in a graph;
they take as input a graph G, an integer k and an augmentation policy P and return a
set S of the best shortcuts for augmenting G. Following, we describe the methods that
we employ in the experimental evaluation. These reflect the case of network-centric
policy. In the case of the node-centric policy, the procedure is similar, but the method
has to be repeated until the best k shortcuts for each node in G have been determined.
The following methods are considered.
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Table II. Networks
Name
#Nodes
#Edges
#CS
#FoFs
Density
L̄(G)
dolphins
netsc
yeast
62
379
2,224
159
914
7,049
1,732
70,717
2,464,927
448
2,916
73,479
0.084
0.012
0.003
3.357
6.042
4.376
ba-s
ba-m
ba-l
50
500
1,000
49
499
999
1,176
124,251
498,501
99
1,685
3,501
0.040
0.004
0.002
4.641
7.398
8.132
comm-s
comm-m
comm-l
50
500
1,000
127
1,469
2,959
1,098
123,281
496,541
197
6,589
14,620
0.103
0.012
0.006
3.786
6.241
6.764
—Greedy-S: The greedy method described in Section 5.1.
—Greedy-B: The batch variant of Greedy-S described in Section 5.2.
—PS: Our path screening method described in Section 5.3.
—PSS: Our sampling-based variant of PS described in Section 5.4.
—Random: A baseline method that selects k shortcuts uniformly at random to add in
G, inspired by Meyerson and Tagiku [2009].
—Mutual Friends Intersection (MFI): This is another sensible baseline method that
selects shortcuts based on the number of adjacent nodes that two nodes share. This
method is inspired by the common practice of online social networking services to
suggest new friendships (shortcuts) to users based on the number of their mutual
friends. If N(x) is the neighborhood of node x and N(y) is the neighborhood of node y,
then the utility score of a shortcut (x, y) can be expressed as the size of the intersection
of the neighborhoods of x and y.
Network Topologies: For the needs of our experimental evaluation, we consider both
real and synthetic network topologies. The real network topologies, represent connected,1 undirected, unweighted networks (dolphins [Lusseau et al. 2003], netsc
[Newman 2006], yeast [Bu et al. 2003]). The synthetic network topologies are of two
types. Networks of the first type simulate a preferential attachment network topology
based on the Barabási–Albert network model [Barabási and Albert 1999]. The number
of nodes and edges in these networks have been selected so as to resemble trees; trees
are typically sparse (less dense) and the shortest paths between pairs of nodes are
unique. We experiment with three of them, a small (ba-s), a medium (ba-m), and a
large (ba-l). Networks of the second type represent tightly knit communities (clusters).
These networks are typically more dense and have the characteristic that for a large
number of pairs of nodes there are many alternative shortest paths that connect them.
We experiment with three of them, a small (comm-s), a medium (comm-m), and a large
(comm-l). Table II provides a summary of the networks we consider.2 For each network,
we report the number of candidate shortcuts, the number of candidate shortcuts that
represent Friends of Friends (FoFs), the network density (Density = 2|E|\(|V |(|V |−1)))
and its characteristic path length L̄(G).
Evaluation Metrics: We assess the performance of the various methods according
to accuracy and efficiency measures. Accuracy concerns the degree of usefulness of
adding a specific set of shortcuts, in terms of optimizing a network property. To assess
1 In
cases where a network is not connected, we extract its largest connected component and operate on it.
the choice of the graph sizes considered, one should bear in mind that in this problem we are
interested in the evaluation of the utility of missing edges (shortcuts) of a graph, which in our cases range
from a few thousands and hundreds of thousands to millions, as is depicted in Table II.
2 Regarding
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M. Papagelis
the accuracy of the various algorithms we define a gain metric, which expresses the
percentage change on the characteristic path length, L̄, in the original graph G and in
the augmented graph Gaug . Formally, the gain is given by:
Gain =
| L̄(G) − L̄(Gaug )|
L̄(G)
× 100.
Efficiency concerns the time performance of a shortcut edge addition method. To assess
the time performance of a method we measure the period of its execution time in
milliseconds.
Computing Environment: All our experiments have been conducted on a Dell XPS
8700 desktop running 64-bit Windows 8.1. The desktop is equipped with an Intel Core
i7-4770 CPU 3.40GHz with 24.0GB of RAM. The Intel Core i7-4770 processor has
four cores and uses hyperthreading, so it appears to have eight logical CPUs. Our
algorithms have been implemented using the Java programming language version 8.0
(build 1.8.0_31) and the experiments are executed on a Java HotSpot 64-Bit Server VM
(build 25.31-b07, mixed mode).
6.1. Evaluation of the Path Screening Method
Our experiments are grouped into three sets. For the methods that introduce randomness or are based on computing shortest path trees (i.e., all methods other than
Greedy-S), single runs are susceptible to introduce bias due to randomness. We alleviate this bias by repeating the experiment many times (x10) and averaging together
the results. Then, we report average values of accuracy and efficiency.
6.1.1. PS Performance/Network-centric Policy. We evaluate the accuracy and efficiency of
our path screening method against the greedy methods, for a varying range of network
topologies and number of edges k to be added. More specifically, we experiment with
dolphins, ba-s, and comm-s and we add 1, 5, and 10 shortcuts. For this set of experiments, we had to limit our experiments on very small networks and on network policy,
because Greedy-S is extremely slow (as we discuss below), so it is not appropriate for
larger graphs or cases of node-centric policy (because the set of candidate shortcuts
that need to be evaluated is very large).
Accuracy: Figure 7(a) presents the accuracy results for the various networks. In all
instances, Greedy-S performs better than any other method. This is because Greedy-S
evaluates any of the k edges one at a time. Even if this method is still suboptimal (due
to making locally optimal choices), it ensures that at each iteration the correct utility
for each of the subsequent candidate shortcuts is computed, taking into consideration
the previously added ones. As such, it avoids adding shortcuts that have overlapping
utility. On the other hand, Random performs poorly in all instances, as expected, and
forms a baseline for the performance of the other methods. The accuracy performance
of the rest two methods, Greedy-B and PS is comparable. They both perform only
slightly worse than Greedy-S, but outperform Random. Overall, the performance of
PS is always comparable and sometimes better than that of the Greedy-B method.
On another note, the difference between the performance of Greedy-S and the other
methods is becoming more evident as k is increasing. This is to be expected; as all the
other methods add shortcuts on a batch way, the larger the number of added shortcuts
is, the larger the likelihood that these shortcuts share overlapping st-shortest paths
will be, rendering the overall gain to be smaller.
Efficiency: Table III presents the efficiency results for the various networks. Note
that, for all methods other than the Greedy-S, it is sufficient to report only one value
for each network, since these methods will evaluate the utility of all shortcuts only
once. On the other hand, for the case of Greedy-S, we report values for all different
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Fig. 7. Accuracy of our path screening method (PS) and its sampling-based variant (PSS).
instantiations of k (i.e., Greedy-S-1, Greedy-S-5, Greedy-S-10). It becomes obvious from
Table III that Greedy-S is extremely inefficient, as it adds one shortcut at a time. In
fact, Greedy-S will be (almost) k times slower than Greedy-B. More importantly, our
path screening method, PS, outperforms Greedy-B by a multitude of times (i.e., x164,
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M. Papagelis
Table III. PS Efficiency (ms) / Network Centric
Method
Greedy-S-1
Greedy-S-5
Greedy-S-10
Greedy-B
PS
Random
dolphins
ba-s
comm-s
7,575
33,495
65,435
7,729
47
1
2,919
10,830
20,295
3,004
39
1
3,793
14,050
27,849
3,849
41
1
Table IV. PS Efficiency (ms) / Node Centric
Method
Greedy-B
PS
Random
dolphins
ba-s
comm-s
7,718
47
1
3,033
31
1
3,829
32
1
x77, and x93 times faster, respectively). The gain in efficiency is due to the fact that,
while Greedy-B computes the utility of all shortcuts, PS looks for shortcuts that connect
nodes that lie on existing shortest paths. As such, our method suggests huge savings
in the computation of shortcut utility.
6.1.2. PS Performance/Node-Centric Policy. We evaluate the accuracy and efficiency of
our path screening method PS against Greedy-B and Random, for a varying range of
network topologies. More specifically, we experiment with dolphins, ba-s, comm-s, and
we ask to add 1, 2, and 3 shortcuts, following the node-centric policy (i.e., for a graph of
n nodes, we ask to find n, 2n, 3n shortcuts, respectively). We had to constrain our tests
to smaller networks due to the inefficiency of the Greedy-B method.
Accuracy: Figure 7(b) presents the accuracy results for the various networks. In
all instances, PS and Greedy-B perform much better than the Random method. It is
also easy to see that as k increases the gain increases as well. This is to be expected;
the larger the number of shortcuts added in the network, the lower is the average
shortest path of the network and therefore, the larger the gain would be. Overall,
the performance of PS is comparable to Greedy-B and always better than that of the
Random algorithms.
Efficiency: Table IV presents the efficiency results for the various networks. Again,
our path screening method, PS, outperforms Greedy-B by a a multitude of times (i.e.,
x164, x97 and x119 times faster, respectively).
6.1.3. PSS Performance. We have demonstrated that PS is extremely faster than
Greedy-S and Greedy-B, while maintaining high levels of accuracy. However, the applicability of PS can be limited by the need to compute all-pairs shortest paths in the
graph. In this set of experiments, we evaluate the accuracy and efficiency of PSS, a
sampling-based variant of our path screening method.
Accuracy: We assess the performance of PSS against the PS method that serves as
the ground truth and the Random that serves as baseline for our sampling method.
We experiment in large networks (yeast, ba-l, comm-l) and for varying k, as well as,
varying sample size. In particular, for each network, we experiment with sample sizes
of 1%, 2%, and 3% of the total nodes in the network that define the methods PSS (1%),
PSS (2%), and PSS (3%), respectively, and at each run we consider adding 1, 2, and
3 shortcuts, following the node-centric policy. Figure 7(c) presents the results for the
various instances. In all cases, the sampling accuracy increases with the sample size.
This is expected, as more shortest paths are evaluated in larger samples. Moreover,
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Table V. PSS Efficiency (ms) / Node Centric Policy
Method
yeast
Random
250
PSS (1%)
490
PSS (2%)
848
PSS (3%) 1,179
PS
16,475
ba-l
comm-l
187
157
344
437
4,365
181
172
282
422
4,764
Table VI. Large Networks
Name
facebook
astro-ph
enron
cond-mat
twitter
dblp
amazon
youtube
#Nodes
#CS
Density
4,039
14,845
33,696
36,458
#Edges
88,234
119,652
180,811
171,736
8.1E+06
1.1E+08
5.6E+08
6.6E+08
1.1E-02
1.1E-03
3.2E-04
2.5E-04
81,306
317,072
334,860
1,134,791
1,342,310
1,049,779
925,838
2,986,629
3.3E+09
5.0E+10
5.6E+10
6.4E+11
4.1E-04
2.1E-05
1.6E-05
4.6E-06
PSS has always a better accuracy than Random and is only slightly lower than that
of PS.
Efficiency: Table V presents the efficiency results for the various methods in varying
networks. The execution time of PS represents the time of the slowest method. For
example, while PS requires around 4.76s for finding shortcuts to add in the comm-l
network, PSS (1%) requires less than 0.2s (i.e., around x27 times faster). At the same
time, the accuracy of PSS (1%) is comparable to that of PS for all k (as depicted in
Figure 7(c)). Overall, PSS can be used to boost the performance of PS, without duly
affecting its accuracy.
6.1.4. PSS Scalability. We have demonstrated that PSS is a multitude of times faster
than standard approaches (Greedy-S and Greedy-B) and the nonsampling version of our
algorithm (PS), while maintaining high levels of accuracy. In this set of experiments,
we evaluate the scalability of PSS in much larger networks, following the node-centric
policy, where k = 1 (i.e., we seek to add a new shortcut to each node). For this set of
experiments, the rest of the algorithms we have discussed cannot be evaluated, as they
are not designed to scale. In particular, we employ a few of the larger network datasets
freely available online,3 including an email communication network (enron), three
coauthoring collaboration networks (astro-ph, cond-mat, dblp), four social networks
(facebook, twitter, youtube) and a network of copurchased products (amazon). Details
about these networks can be found in Table VI and in Leskovec and Krevl [2014]. To
allow our PSS algorithm to scale, we control the sample size; for the smaller of these
graphs, we considered a sample of 1% of the network nodes [PSS (1%)] and for the
larger of these graphs we considered a sample of 0.01% of the network nodes [PSS
(0.01%)]. Recall that, as we discussed in Section 5.4, each of the nodes of the sample
is used as a reference node (or root) on which we execute the Dijkstra algorithm and
compute the single source shortest path tree from that node to all other nodes in the
graph. Then, these shortest path trees are processed in order to determine the best
shortcuts to add in the graph. Table VI also lists the number of all candidate shortcuts
in the graph and the density of each of these large networks. It is important to note
3 snap.stanford.edu/data/.
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M. Papagelis
Table VII. PSS Scalability (ms)
Method
PSS (1%)
facebook
astro-ph
enron
cond-mat
2,563
26,002
114,360
175,620
Table VIII. PSS Scalability (ms)
Method
twitter
dblp
amazon
youtube
PSS (0.01%)
12,845
247,829
643,749
2,976,239
that the number of candidates shortcuts can be really big; in the case of the youtube
network, this is approximately 640 billion candidate shortcuts.
Efficiency: Tables VII and VIII present the time efficiency results for the PSS (1%) and
the PSS (0.01%) method, respectively, for a number of large networks. It is clear that
our PSS method can scale to large networks very well, while standard methods are not
even applicable. For example, PSS (0.01%) needs less than 50min (i.e., 2,976,239 ms) to
find which shortcuts to add in the youtube network that consists of ∼1.1 million nodes,
∼3 million edges and a total of ∼640 billion candidate shortcuts. Overall, PSS can be
used to boost the performance of PS.
Accuracy: Accuracy results for this set of experiments are excluded, as they would
require to report on the performance of our algorithms against alternatives methods.
But, as discussed earlier, alternative methods do not scale well and therefore it is not
possible to apply them on networks of this size.
6.2. The Cascade Effect
We have demonstrated that our path screening method outperforms the accuracy of
sensible baselines and at the same time is extremely efficient. When we introduced
the problem we made a connection between the efficiency of a network to propagate
information and its network structure. In this set of experiments, we evaluate the
impact of our methods in the cascade process. In particular, we evaluate the impact
on the graph conductance, on the Markov Chain mixing time and on the information
propagation spread.
6.2.1. Impact on Graph Conductance. We evaluate the impact of our methods in increasing the conductance of a graph. More specifically, we experiment with both real and
synthetic networks of small, medium, and large size. For each case, we add 1, 5, or 10
shortcuts using the Random, PS, or Greedy-B method, following the network-centric
augmentation policy. Figure 8 shows the effect of each method on the original graph’s
conductance. Note that we had to constrain the computations of the Greedy-B method
to only small networks due to its inefficiency. The plots indicate that the graph conductance (minimum conductance of any cluster size) is increased more when we augment
the graph using our path screening method PS, than the Random baseline and
(almost) always this increase is only slightly smaller than that achieved by Greedy-B.
Furthermore, conductance is increasing with k; this is depicted by the larger increase
in the cases of adding 5 or 10 shortcuts, for all networks. Moreover, this increase is
more expressed in the case of the community networks comm-s, comm-m, comm-l.
This is to be expected, as these networks consist of only a few cross-cutting edges.
6.2.2. Impact on Markov Chain Mixing Time. We evaluate the impact of our methods
in decreasing the mixing time of a graph using the same parameter settings as in
Section 6.2.1. We report the theoretical mixing time of the fastest random walk in a
network, in terms of the number of steps (i.e., random walk length) before it converges
(i.e., reaching a specific relative point-wise threshold ). For the needs of our experiments, we set = 0.01. It is easy to see that a smaller threshold would end up in
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Fig. 8. Impact on graph conductance on small (top), medium (middle), and large (bottom) graphs.
larger savings and a larger one in smaller savings. Figure 9 shows the effect of each
method on the original graph’s mixing time. The plots indicate that the mixing time
is decreasing more when we augment the graph using our path screening method PS,
than the Random baseline and (almost) always this decrease is comparable to the one
caused by Greedy-B. The trend is consistent for yeast, but does not appear properly
due to small values. Furthermore, the mixing time is decreasing with the number of
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M. Papagelis
Fig. 9. Impact on Markov chain mixing time on small (top), medium (middle), and large (bottom) graphs.
shortcuts added and this decrease is, as with the graph conductance, more expressed
in the case of the community-like networks.
6.2.3. Impact on Information Propagation Spread. We have demonstrated that our method
can dramatically affect properties of the graph that are critical for characterizing its
ability to carry on propagation processes, such as the graph conductance and its mixing
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time. In this set of experiments, we evaluate the impact of our methods on the cascade
spread; the number of nodes reached by a propagation process. It is important to
note that our methods are generic and do not depend on a specific propagation model.
Consequently, we first present a simple generic propagation model that is used in the
evaluation and then we describe the evaluation in detail.
Propagation Model: We employ a basic threshold propagation algorithm based on
an individual-level natural model of direct-benefit effects in networks due to Stephen
Morris [Morris 2000]. The underlying consideration of this model is that a node has a
certain social network neighbors and the benefits to adopt a new behavior increase as
more and more of these neighbors adopt it. Conforming to the model, a set of initiator
nodes I in the network has adopted a behavior. At each round, a node should adopt
the new behavior once a sufficient proportion of its neighbors have done so. We model
the decision making using a simple threshold rule; if a fraction of at least T of a
node’s neighbors have adopted, then it should, too. The semantics of the threshold are
intuitive; the smaller the T is the more attractive the cascading behavior is, as you
need fewer of your neighbors to engage before you do. Nodes that have already adopted
the behavior cannot switch back. The propagation process terminates when no change
is detected in a specific round (no new node has adopted). In a sense, the number of
nodes that have adopted in the end of the propagation process represents a sphere of
influence (SOI), of the initiator set I.
Given a graph, first we add new edges based on our path screening method, PS, and
then add new edges based on MFI. For the needs of this experiment, we had to constrain
the set of candidate shortcuts to represent FoFs (CS = {(x, y) : R(x, y) = 2}). We
experiment with three medium size graphs, netsc, ba-m, and comm-m, and ask to add
one shortcut to each node (node-centric policy, k = 1) for variable sizes of the initiator
set I and values of the threshold T . For each case, we simulate the propagation model
on the augmented network several times (x100), each time computing the number of
nodes that have adopted, and averaging over all instances. In the end, we report the
average size of the sphere of influence, SOI(%), as percentage of the total number of
nodes in the network for finer representation:
|SOI|
SOI(%) =
.
|V |
Figure 10 presents the results for the various networks. In all instances, PS outpeforms
MFI. Note that the spheres of influence of the various initiator sets I (2%, 4%, 6%, 8%,
10% of the total number of nodes) in networks augmented by our PS method, are always
larger than or equal to the ones in networks augmented by MFI—an increase that
ranges between 0% and ∼80%. This behavior is demonstrated in all three networks,
but the impact is larger in comm-m and netsc; the two networks that have a more
profound clustering structure. On another note, as T varies (0.2, 0.3, 0.4), the sizes
of the spheres of influence are getting smaller. This is expected as larger threshold
indicates a less attractive behavior that is difficult to be adopted by nodes, so irrelevant
of the structure of the augmented network, the behavior cannot easily cascade.
7. RELATED WORK
We have tried to provide pointers to work related to our research throughout the paper.
In this Section, we provide a more concrete coverage of related work not mentioned earlier. In particular, the literature related to topics of graph augmentation, information
propagation, and link prediction and recommendation.
7.1. Graph Augmentation
In traditional graph augmentation problems, we ask to find a minimum-cost set
of edges to add to a graph to satisfy a specified property, such as biconnectivity,
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M. Papagelis
Fig. 10. Impact on information propagation spread.
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bridge-connectivity or strong connectivity. These augmentation problems have been
shown to be NP-complete in the restricted case of connected graphs [Eswaran and Tarjan 1976], while a number of approximation algorithms with favorable time complexity
have been shown to have constant worst-case performance ratios [Frederickson and
JáJá 1981; Khuller and Thurimella 1993; Nutov 2005]. In one of the first works that
considers the problem with a hard limit on the number of edges to be added, Meyerson and Tagiku [Meyerson and Tagiku 2009] considered the problem of minimizing the
average distance between the nodes. They obtained several constant-factor approximations using the k-median with penalties problem. They also improved the best known
approximation ratio for metric k-median with penalties, to obtain better approximation factors for the other problems they considered. If α denotes the best approximation
known for metric k-median with penalties, they presented an α-approximation for the
single-source average-shortest-path problem, and a 2α-approximation for the general
average shortest-path problem. In an another recent work, Demain and Morteza [Demaine and Zadimoghaddam 2010] studied the problem of minimizing the diameter
of a graph by adding k shortcut edges, for speeding up communication in an existing
network design. They developed constant-factor approximation algorithms for different variations of the problem and showed how to improve the approximation ratios
using resource augmentation to allow more than k shortcut edges. Both [Meyerson
and Tagiku 2009] and [Demaine and Zadimoghaddam 2010] observe a close relation
between the single-source version of the problem, in which we want to minimize the
largest distance from a given source vertex, and the well-known k-median problem. In
the single-source version of the problem, a node may want to construct edges in order to
minimize its distance from the other nodes. Laoutaris et al. [2008] studied the bounded
budget connection game, where nodes in a network have a budget for purchasing links
with the objective to minimize their average distance to the other nodes. A version of
the problem for minimizing the maximum distance to the other nodes was considered
as well. In a similar context, Zhou et al. [2013] develop network engineering algorithms
that aim to provide performance gains of random walk simulations over networks by
reducing their mixing time.
The problem of interest in this paper, is not the same as the aforementioned research.
In the traditional graph theory, connectivity (biconnectivity, triconnectivity, etc.) of a
finite undirected graph refers to the minimum number of disjoint paths that can be
found between any pair of vertices, as characterized by Menger’s theorem [Menger
1927] and generalized by the max-flow min-cut theorem [Papadimitriou and Steiglitz
1998]. In our research, the focus is on bringing nodes closer to each other, a connectivity
property that is better expressed as the characteristic path length of a graph. It is also
different to the theoretical approach of Meyerson and Tagiku [2009] as the suggested
method operates by always attaching shortcuts to a single node, leading to a star
network topology; in our problem, we need to be able to control where shortcuts are
added in the network. In the rest of the approaches, variations of the problem of interest
are considered, where either the context of the nodes needs to be known (assumed) or
the emphasis is on optimizing network properties different to its characteristic path.
Our research stems from and extends on Papagelis et al. [2011], where the problem
of adding k shortcuts to a graph in order to minimize its characteristic path length was
originally introduced. Interesting variations of the problem exist, as well. In Lazaridou
et al. [2015], the authors consider the problem of identifying the set of the top-k pairs
of nodes in a graph whose distances are reduced the most as the result of adding new
edges. In Parotisidis et al. [2015], the authors suggest methods for solving the problem
when a more manageable subset of the candidate edges is considered (sublinear in the
size of the graph), while the candidate set in our problem consists of all nonexistent
edges (quadratic in the size of the graph).
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M. Papagelis
7.2. Maximizing Information Propagation
The idea of making structural changes in a network to enable information to travel
easier from a node to another is closely related to the idea of finding influential nodes
in a network to maximize information spread [Chaoji et al. 2012; Kempe et al. 2003;
Kleinberg 2007; Richardson and Domingos 2002] or detecting outbreaks in a network
[Leskovec et al. 2007]. Manipulating edges to control information spread is also the
topic in [Tong et al. 2012]. In the work of Richardson and Domingos [Richardson and
Domingos 2002], the problem of finding the most influential set of k nodes in a network was introduced, motivated by viral marketing. Viral marketing refers to marketing techniques that aim to increase a marketing objective (such as brand awareness)
through replicating an epidemic process in social networks, similar to the diffusion
of innovations. It is useful in applications that employ the diffusion process, such as
maximizing the spread of influence in a social network [Kempe et al. 2003] and early
detection of outbreaks in a network [Leskovec et al. 2007].
The problem of interest in this paper, is not the same as the problem of finding
influential nodes in the network, as we focus on importance of edges and not nodes.
More importantly, we do not make any assumption about the context of the nodes or
factors that can affect an information propagation process or model (e.g., set of initiators
or early adopters, how influential is a node to its neighbors, how susceptible is a node
to its neighbors’ influence and so on). On the contrary, our problem tries to improve on
information propagation processes, by operating directly on the network structure.
7.3. Link Prediction and Recommendation
Our problem draws connections to the problem of link prediction [Backstrom and
Leskovec 2011; Liben-Nowell and Kleinberg 2003; Popescul et al. 2003], in which the
objective is to predict which new interactions among members of a social network are
likely to occur in the near future. Backstrom and Leskovec [2011] study the problem
of link prediction and recommendation in social networks by performing supervised
random walks on the network. Their method combines network structural information
with rich node and edge attribute data to guide random walks. In a similar context,
Tian et al. [2010] study the link revival problem, where the objective is to turn already
existing edges with a few interactions to be more active so that the resulted connection
will improve the social network connectivity. The authors develop an algorithm that
explores local properties of a an evolving graph and try to recommend links per node
at a time. The basic idea of their algorithm is to first consider graph snapshots in
distinct time intervals, by monitoring the social interaction graph. Then, predict the
probability with which a social interaction will happen in the future and if it has a
low probability to happen, then they try to estimate the gain in the connectivity by
adding this edge and consider it for recommendation. Finally, they recommend edges
that have the larger gain in connectivity.
The problem of interest in this paper, is not the same as the link prediction and
recommendation problems described previously. We are not interested in using historical data of interactions to predict future links or to revive existing but weak links.
Instead, we want to suggest new edges that minimize global structural properties of
the network by operating solely on the network structure.
8. CONCLUSIONS
We considered the problem of adding a small number of shortcuts to a graph in order
to optimize a connectivity property and improve its capacity of carrying on social
processes. This is a quite novel, interesting, and challenging problem. We proposed a
novel method for quickly evaluating the importance of candidate shortcuts in a graph.
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More specifically, our method is able to approximate the number of shortest paths that
would run along a candidate shortcut, if it was already in the network. Intuitively, our
method approximates the betweenness centrality of these nonexisting edges [Newman
2005]. Betweenness centrality of an edge or simply edge betweenness is the number
of shortest paths between pairs of nodes that run along it and is a measure of the
influence of an edge over the flow of information among nodes, especially in cases where
information flow over a network primarily follows the shortest available path [Girvan
and Newman 2002; Newman 2005]. We proposed, as well, a sampling-based variant of
the method that can be used to scale up the computation for larger graphs. Through
experiments on various datasets, we demonstrated that our approach outperforms
sensible baselines in both accuracy and efficiency, for a varying range of conditions.
Overall, the algorithms we described are simple to understand and implement, accurate, very fast, and general, so they can probably be easily adopted in a variety of
strategies. As such, we expect our methods to be beneficial in diverse settings and disciplines, ranging from social to technological networks. Later, we discuss a few ideas
related to our research that aim to enlarge or prolong its scope. To some degree they
present interesting extensions to our work and offer alternative views and insights.
8.1. Data Parallelism
It is important to note that further speedup of our methods can be achieved by technologies that distribute data and computing tasks across different parallel computing
nodes. Data parallelism is based on distributing the data and computing tasks across
different parallel computing nodes.
The first component of our main algorithm, based on a variation of Johnson’s algorithm, is easy to distribute because it runs Dijkstra’s algorithm n times, one for
each node, which can be done in parallel. For example, all-pairs shortest paths can
be computed by employing distributed versions of breadth-first or depth-first algorithms [Awerbuch 1985; Awerbuch and Gallager 1985; Chandy and Misra 1982]. New
programming models for processing large datasets with a parallel, distributed algorithm on a cluster, such as MapReduce [Dean and Ghemawat 2004] and Hadoop
[Shvachko et al. 2010], can be used, as well, to compute the all-pairs shortest paths
component of our main algorithm: the Map() procedure performs multiple parallel
breadth-first searches simultaneously assuming source nodes {s0 , s1 , . . . , sn} and instead of emitting a single distance, emits an array of distances with respect to each
source, as well as, information about the nodes that constitute each path. Then, the Reduce() procedure selects a path with minimum length for each element from the array.
The second component of our main algorithm is to process the list of shortest paths
(path screening) in parallel in order to compute the utility of each candidate shortcut.
Algorithm 3 and Algorithm 4 show how to employ MapReduce in this step, as well: the
Map() (Algorithm 3) performs the path screening of each shortest path in isolation and
computes the utility of any candidate shortcut that occurs in the current path. Then,
the Reduce() (Algorithm 4) performs a summary operation that computes the utility
scores of candidate shortcuts among all shortest paths. We do not further elaborate on
this issue due to lack of space and as it is orthogonal to our problem.
8.2. Disconnected Networks
We have constrained our discussion to connected networks. In the case of disconnected
networks, one should first consider finding the connected components of the input
network and then connect them with each other. Finding a smallest augmentation that
connects an undirected graph is a well-known problem, this is the problem of finding a
spanning tree, with many efficient algorithms available.
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12:32
M. Papagelis
ALGORITHM 3: Data Parallelism Using MapReduce − Map()
Input: An integer i representing a batch of shortest paths
Output: Partial utility of any candidate shortcut found in the ith batch
for each shortest path in the ith batch do
(x, y) ←− a candidate shortcut that occurs in a path;
δ ←− the range of the shortcut in G (δ = R(x, y));
produce one output record < (x, y), δ >;
end
ALGORITHM 4: Data Parallelism Using MapReduce − Reduce()
Input: Set of candidate shortcuts (x, y)
Output: The utility of each candidate shortcut
for each candidate shortcut < (x, y), δ > do
accumulate in U the sum of δ;
end
produce one output record < (x, y), U >;
8.3. Friend Suggestion Implications
The paper also highlighted the fact that the global optimization problem of augmenting
a graph by adding new edges can be seen as an alternative for friend suggestion in a
social network. The premise is that in the augmented network users are more well
connected and information can spread more efficiently in the network. In this Section,
we detail a methodology to evaluate such an alternative and relevant implications.
Current state-of-the-art friend suggestion algorithms are mainly based on the idea of
suggesting friends with whom an individual shares a large number of mutual friends
(similar to the MFI algorithm we presented in Section 6); that way the social network
tries to maximize the probability of a new connection to occur (be realized). However,
from a graph perspective, such an algorithm operates only locally by making the local network (cluster) more dense, but does not succeed in optimizing global graph
properties (such as better information propagation). An alternative friend suggestion
algorithm would try to blend ideas of our graph augmentation algorithms and the
current-state-of-the-art by suggesting a set of mixed friends including:
—friends that make the local network more dense (coming from current state-of-the-art
friend suggest algorithms), and
—friends that try to optimize global network properties (coming from our graph augmentation algorithms).
It becomes clear that many hybrid strategies can be followed to combine and evaluate
the aforementioned algorithms. The evaluation should seek to optimize on the number
of friends suggested by each algorithm and on the best way to rank the suggested
friendships before presenting them to the end user. We do not further elaborate on this
issue due to lack of space and as it is orthogonal to our problem.
Another important aspect of the friend suggest problem is that while we suggest
friends, we cannot be sure that these friendships will be really materialized (i.e., accepted by the users). Therefore, we have to reason in probabilistic terms. The probability model is based on the assumption that for each candidate shortcut (x, y), there
is an associated probability p(x, y) for it to be realized once suggested. The probability
p can be given by any simple prediction method (e.g., it can be based on the number of
ACM Transactions on Knowledge Discovery from Data, Vol. 10, No. 2, Article 12, Publication date: October 2015.
Refining Social Graph Connectivity via Shortcut Edge Addition
12:33
mutual friends between two users). Then, the objective is to suggest a set of edges that
will maximize the expected utility, if added in the graph.
ACKNOWLEDGMENTS
We are thankful to Francesco Bonchi and Aristides Gionis for thoughtful and stimulating discussions and
Nick Koudas for providing constructive feedback and proof-reading draft versions of the manuscript.
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Received May 2014; revised February 2015; accepted April 2015
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